Theorem cfinufil 23882

Description: An ultrafilter is free iff it contains the Fréchet filter cfinfil 23847 as a subset. (Contributed by NM, 14-Jul-2008.) (Revised by Stefan O'Rear, 2-Aug-2015.)

Assertion

Ref	Expression
cfinufil	⊢ (𝐹 ∈ (UFil‘𝑋) → (∩ 𝐹 = ∅ ↔ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑥) ∈ Fin} ⊆ 𝐹))

Distinct variable groups:   𝑥,𝐹   𝑥,𝑋

Proof of Theorem cfinufil

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elpwi 4587	. . . . 5 ⊢ (𝑥 ∈ 𝒫 𝑋 → 𝑥 ⊆ 𝑋)
2		ufilb 23860	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ⊆ 𝑋) → (¬ 𝑥 ∈ 𝐹 ↔ (𝑋 ∖ 𝑥) ∈ 𝐹))
3	2	adantr 480	. . . . . . . . 9 ⊢ (((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ⊆ 𝑋) ∧ (𝑋 ∖ 𝑥) ∈ Fin) → (¬ 𝑥 ∈ 𝐹 ↔ (𝑋 ∖ 𝑥) ∈ 𝐹))
4		ufilfil 23858	. . . . . . . . . . . 12 ⊢ (𝐹 ∈ (UFil‘𝑋) → 𝐹 ∈ (Fil‘𝑋))
5	4	adantr 480	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ⊆ 𝑋) → 𝐹 ∈ (Fil‘𝑋))
6		filfinnfr 23831	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑋 ∖ 𝑥) ∈ 𝐹 ∧ (𝑋 ∖ 𝑥) ∈ Fin) → ∩ 𝐹 ≠ ∅)
7	6	3exp 1119	. . . . . . . . . . . 12 ⊢ (𝐹 ∈ (Fil‘𝑋) → ((𝑋 ∖ 𝑥) ∈ 𝐹 → ((𝑋 ∖ 𝑥) ∈ Fin → ∩ 𝐹 ≠ ∅)))
8	7	com23 86	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (Fil‘𝑋) → ((𝑋 ∖ 𝑥) ∈ Fin → ((𝑋 ∖ 𝑥) ∈ 𝐹 → ∩ 𝐹 ≠ ∅)))
9	5, 8	syl 17	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ⊆ 𝑋) → ((𝑋 ∖ 𝑥) ∈ Fin → ((𝑋 ∖ 𝑥) ∈ 𝐹 → ∩ 𝐹 ≠ ∅)))
10	9	imp 406	. . . . . . . . 9 ⊢ (((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ⊆ 𝑋) ∧ (𝑋 ∖ 𝑥) ∈ Fin) → ((𝑋 ∖ 𝑥) ∈ 𝐹 → ∩ 𝐹 ≠ ∅))
11	3, 10	sylbid 240	. . . . . . . 8 ⊢ (((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ⊆ 𝑋) ∧ (𝑋 ∖ 𝑥) ∈ Fin) → (¬ 𝑥 ∈ 𝐹 → ∩ 𝐹 ≠ ∅))
12	11	necon4bd 2951	. . . . . . 7 ⊢ (((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ⊆ 𝑋) ∧ (𝑋 ∖ 𝑥) ∈ Fin) → (∩ 𝐹 = ∅ → 𝑥 ∈ 𝐹))
13	12	ex 412	. . . . . 6 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ⊆ 𝑋) → ((𝑋 ∖ 𝑥) ∈ Fin → (∩ 𝐹 = ∅ → 𝑥 ∈ 𝐹)))
14	13	com23 86	. . . . 5 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ⊆ 𝑋) → (∩ 𝐹 = ∅ → ((𝑋 ∖ 𝑥) ∈ Fin → 𝑥 ∈ 𝐹)))
15	1, 14	sylan2 593	. . . 4 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ∈ 𝒫 𝑋) → (∩ 𝐹 = ∅ → ((𝑋 ∖ 𝑥) ∈ Fin → 𝑥 ∈ 𝐹)))
16	15	ralrimdva 3141	. . 3 ⊢ (𝐹 ∈ (UFil‘𝑋) → (∩ 𝐹 = ∅ → ∀𝑥 ∈ 𝒫 𝑋((𝑋 ∖ 𝑥) ∈ Fin → 𝑥 ∈ 𝐹)))
17	4	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑦 ∈ ∩ 𝐹) → 𝐹 ∈ (Fil‘𝑋))
18		uffixsn 23879	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑦 ∈ ∩ 𝐹) → {𝑦} ∈ 𝐹)
19		filelss 23806	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ {𝑦} ∈ 𝐹) → {𝑦} ⊆ 𝑋)
20	17, 18, 19	syl2anc 584	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑦 ∈ ∩ 𝐹) → {𝑦} ⊆ 𝑋)
21		dfss4 4249	. . . . . . . . . . 11 ⊢ ({𝑦} ⊆ 𝑋 ↔ (𝑋 ∖ (𝑋 ∖ {𝑦})) = {𝑦})
22	20, 21	sylib 218	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑦 ∈ ∩ 𝐹) → (𝑋 ∖ (𝑋 ∖ {𝑦})) = {𝑦})
23		snfi 9065	. . . . . . . . . 10 ⊢ {𝑦} ∈ Fin
24	22, 23	eqeltrdi 2841	. . . . . . . . 9 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑦 ∈ ∩ 𝐹) → (𝑋 ∖ (𝑋 ∖ {𝑦})) ∈ Fin)
25		difss 4116	. . . . . . . . . . 11 ⊢ (𝑋 ∖ {𝑦}) ⊆ 𝑋
26		filtop 23809	. . . . . . . . . . . 12 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝑋 ∈ 𝐹)
27		elpw2g 5313	. . . . . . . . . . . 12 ⊢ (𝑋 ∈ 𝐹 → ((𝑋 ∖ {𝑦}) ∈ 𝒫 𝑋 ↔ (𝑋 ∖ {𝑦}) ⊆ 𝑋))
28	17, 26, 27	3syl 18	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑦 ∈ ∩ 𝐹) → ((𝑋 ∖ {𝑦}) ∈ 𝒫 𝑋 ↔ (𝑋 ∖ {𝑦}) ⊆ 𝑋))
29	25, 28	mpbiri 258	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑦 ∈ ∩ 𝐹) → (𝑋 ∖ {𝑦}) ∈ 𝒫 𝑋)
30		difeq2 4100	. . . . . . . . . . . . 13 ⊢ (𝑥 = (𝑋 ∖ {𝑦}) → (𝑋 ∖ 𝑥) = (𝑋 ∖ (𝑋 ∖ {𝑦})))
31	30	eleq1d 2818	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝑋 ∖ {𝑦}) → ((𝑋 ∖ 𝑥) ∈ Fin ↔ (𝑋 ∖ (𝑋 ∖ {𝑦})) ∈ Fin))
32		eleq1 2821	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝑋 ∖ {𝑦}) → (𝑥 ∈ 𝐹 ↔ (𝑋 ∖ {𝑦}) ∈ 𝐹))
33	31, 32	imbi12d 344	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑋 ∖ {𝑦}) → (((𝑋 ∖ 𝑥) ∈ Fin → 𝑥 ∈ 𝐹) ↔ ((𝑋 ∖ (𝑋 ∖ {𝑦})) ∈ Fin → (𝑋 ∖ {𝑦}) ∈ 𝐹)))
34	33	rspcv 3601	. . . . . . . . . 10 ⊢ ((𝑋 ∖ {𝑦}) ∈ 𝒫 𝑋 → (∀𝑥 ∈ 𝒫 𝑋((𝑋 ∖ 𝑥) ∈ Fin → 𝑥 ∈ 𝐹) → ((𝑋 ∖ (𝑋 ∖ {𝑦})) ∈ Fin → (𝑋 ∖ {𝑦}) ∈ 𝐹)))
35	29, 34	syl 17	. . . . . . . . 9 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑦 ∈ ∩ 𝐹) → (∀𝑥 ∈ 𝒫 𝑋((𝑋 ∖ 𝑥) ∈ Fin → 𝑥 ∈ 𝐹) → ((𝑋 ∖ (𝑋 ∖ {𝑦})) ∈ Fin → (𝑋 ∖ {𝑦}) ∈ 𝐹)))
36	24, 35	mpid 44	. . . . . . . 8 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑦 ∈ ∩ 𝐹) → (∀𝑥 ∈ 𝒫 𝑋((𝑋 ∖ 𝑥) ∈ Fin → 𝑥 ∈ 𝐹) → (𝑋 ∖ {𝑦}) ∈ 𝐹))
37		ufilb 23860	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ {𝑦} ⊆ 𝑋) → (¬ {𝑦} ∈ 𝐹 ↔ (𝑋 ∖ {𝑦}) ∈ 𝐹))
38	20, 37	syldan 591	. . . . . . . . 9 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑦 ∈ ∩ 𝐹) → (¬ {𝑦} ∈ 𝐹 ↔ (𝑋 ∖ {𝑦}) ∈ 𝐹))
39	18	pm2.24d 151	. . . . . . . . 9 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑦 ∈ ∩ 𝐹) → (¬ {𝑦} ∈ 𝐹 → ¬ 𝑦 ∈ ∩ 𝐹))
40	38, 39	sylbird 260	. . . . . . . 8 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑦 ∈ ∩ 𝐹) → ((𝑋 ∖ {𝑦}) ∈ 𝐹 → ¬ 𝑦 ∈ ∩ 𝐹))
41	36, 40	syld 47	. . . . . . 7 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑦 ∈ ∩ 𝐹) → (∀𝑥 ∈ 𝒫 𝑋((𝑋 ∖ 𝑥) ∈ Fin → 𝑥 ∈ 𝐹) → ¬ 𝑦 ∈ ∩ 𝐹))
42	41	impancom 451	. . . . . 6 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ ∀𝑥 ∈ 𝒫 𝑋((𝑋 ∖ 𝑥) ∈ Fin → 𝑥 ∈ 𝐹)) → (𝑦 ∈ ∩ 𝐹 → ¬ 𝑦 ∈ ∩ 𝐹))
43	42	pm2.01d 190	. . . . 5 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ ∀𝑥 ∈ 𝒫 𝑋((𝑋 ∖ 𝑥) ∈ Fin → 𝑥 ∈ 𝐹)) → ¬ 𝑦 ∈ ∩ 𝐹)
44	43	eq0rdv 4387	. . . 4 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ ∀𝑥 ∈ 𝒫 𝑋((𝑋 ∖ 𝑥) ∈ Fin → 𝑥 ∈ 𝐹)) → ∩ 𝐹 = ∅)
45	44	ex 412	. . 3 ⊢ (𝐹 ∈ (UFil‘𝑋) → (∀𝑥 ∈ 𝒫 𝑋((𝑋 ∖ 𝑥) ∈ Fin → 𝑥 ∈ 𝐹) → ∩ 𝐹 = ∅))
46	16, 45	impbid 212	. 2 ⊢ (𝐹 ∈ (UFil‘𝑋) → (∩ 𝐹 = ∅ ↔ ∀𝑥 ∈ 𝒫 𝑋((𝑋 ∖ 𝑥) ∈ Fin → 𝑥 ∈ 𝐹)))
47		rabss 4052	. 2 ⊢ ({𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑥) ∈ Fin} ⊆ 𝐹 ↔ ∀𝑥 ∈ 𝒫 𝑋((𝑋 ∖ 𝑥) ∈ Fin → 𝑥 ∈ 𝐹))
48	46, 47	bitr4di 289	1 ⊢ (𝐹 ∈ (UFil‘𝑋) → (∩ 𝐹 = ∅ ↔ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑥) ∈ Fin} ⊆ 𝐹))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1539   ∈ wcel 2107   ≠ wne 2931  ∀wral 3050  {crab 3419   ∖ cdif 3928   ⊆ wss 3931  ∅c0 4313  𝒫 cpw 4580  {csn 4606  ∩ cint 4926  ‘cfv 6541  Fincfn 8967  Filcfil 23799  UFilcufil 23853

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-sep 5276  ax-nul 5286  ax-pow 5345  ax-pr 5412  ax-un 7737

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-nel 3036  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3420  df-v 3465  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4888  df-int 4927  df-br 5124  df-opab 5186  df-mpt 5206  df-tr 5240  df-id 5558  df-eprel 5564  df-po 5572  df-so 5573  df-fr 5617  df-we 5619  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-rn 5676  df-res 5677  df-ima 5678  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-ov 7416  df-oprab 7417  df-mpo 7418  df-om 7870  df-1o 8488  df-en 8968  df-dom 8969  df-sdom 8970  df-fin 8971  df-fbas 21323  df-fg 21324  df-fil 23800  df-ufil 23855

This theorem is referenced by: (None)